Let $C,D$ be 2-categories and $F,G:C\to D$ be functors. An icon $\alpha:F\to G$ consists of the following:
If $D$ is a strict 2-category (or at least strictly unital), then an icon is identical to an oplax natural transformation whose 1-cell components are identities. In general, there is a bijection between icons and such oplax natural transformations, obtained by pre- and post-composing with the unit constraints of $D$. The name “icon” derives from this correspondence: it is an Identity Component Oplax Natural-transformation.
Icons have technical importance in the theory of 2-categories. For instance, there is no 2-category (or even 3-category) of 2-categories, functors, and lax or oplax transformations (even with modifications), but there is a 2-category of 2-categories, functors, and icons. (In fact, this 2-category is the 2-category of algebras for a certain 2-monad.)
Additionally, if monoidal categories are regarded as one-object 2-categories, then monoidal functors can be identified with 2-functors, and monoidal transformations can be identified with icons.
Icons are also used to construct distributors in the context of enriched bicategories.
Stephen Lack, Icons. Appl. Categ. Structures. Volume 18, Issue 3,(2010), pp 289–307. arxiv:0711.4657
Richard Garner, Mike Shulman, Section 3.8 of Enriched categories as a free cocompletion, Adv. Math. 289 (2016), 1–94
Last revised on July 4, 2017 at 11:32:03. See the history of this page for a list of all contributions to it.